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equal angles
jhz   2
N an hour ago by YaoAOPS
Source: 2025 CTST P16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
2 replies
jhz
4 hours ago
YaoAOPS
an hour ago
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Flee Jumping on Number Line
utkarshgupta   23
N an hour ago by Ilikeminecraft
Source: All Russian Olympiad 2015 11.5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
23 replies
utkarshgupta
Dec 11, 2015
Ilikeminecraft
an hour ago
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Smallest value of |253^m - 40^n|
MS_Kekas   3
N 2 hours ago by imagien_bad
Source: Kyiv City MO 2024 Round 1, Problem 9.5
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
3 replies
MS_Kekas
Jan 28, 2024
imagien_bad
2 hours ago
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Operating on lamps in a circle
anantmudgal09   7
N 2 hours ago by hectorleo123
Source: India Practice TST 2017 D2 P3
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule:

(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is off right now. (Indices taken mod $n$.)

(b) Otherwise, $L_i$ is on right now.

Initially, all the lamps are off, except for $L_1$ which is on. Prove that for infinitely many integers $n$ all the lamps will be off eventually, after a finite amount of time.
7 replies
anantmudgal09
Dec 9, 2017
hectorleo123
2 hours ago
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2025 Caucasus MO Seniors P1
BR1F1SZ   3
N 2 hours ago by Mathdreams
Source: Caucasus MO
For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$[list=a]
[*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation.
[*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation.
[/list]
(Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)
3 replies
BR1F1SZ
4 hours ago
Mathdreams
2 hours ago
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IMO 2018 Problem 2
juckter   95
N 2 hours ago by Marcus_Zhang
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
95 replies
juckter
Jul 9, 2018
Marcus_Zhang
2 hours ago
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Long condition for the beginning
wassupevery1   2
N 2 hours ago by wassupevery1
Source: 2025 Vietnam IMO TST - Problem 1
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$holds for all positive rational numbers $x, y$.
2 replies
wassupevery1
Yesterday at 1:49 PM
wassupevery1
2 hours ago
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Inspired by IMO 1984
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +24abc\leq\frac{81}{64}$$Equality holds when $a=b=\frac{3}{8},c=\frac{1}{4}.$
$$a^2+b^2+ ab +18abc\leq\frac{343}{324}$$Equality holds when $a=b=\frac{7}{18},c=\frac{2}{9}.$
0 replies
sqing
2 hours ago
0 replies
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Prime-related integers [CMO 2018 - P3]
Amir Hossein   15
N 3 hours ago by Ilikeminecraft
Source: 2018 Canadian Mathematical Olympiad - P3
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.

Note that $1$ and $n$ are included as divisors.
15 replies
Amir Hossein
Mar 31, 2018
Ilikeminecraft
3 hours ago
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Inspired by IMO 1984
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +17abc\leq\frac{8000}{7803}$$$$a^2+b^2+ ab +\frac{163}{10}abc\leq\frac{7189057}{7173630}$$$$a^2+b^2+ ab +16.23442238abc\le1$$
2 replies
sqing
Yesterday at 3:04 PM
sqing
3 hours ago
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Inequality with x, y
bel.jad5   6
N Mar 20, 2025 by sqing
Source: Own
Let x and y positive real numbers such that: $x^2+y^2+xy=3$. Find the maximum of $x^2y$
6 replies
bel.jad5
Sep 18, 2016
sqing
Mar 20, 2025
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Inequality with x, y
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inequalities
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Source: Own
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bel.jad5
3750 posts
bel.jad5
#1 h  Sep 18, 2016, 6:30 AM • 2 Y
Y by Adventure10, Mango247
Let x and y positive real numbers such that: $x^2+y^2+xy=3$. Find the maximum of $x^2y$
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  Sep 19, 2016, 4:52 PM • 1 Y
Y by Adventure10
bel.jad5 wrote:
Let x and y positive real numbers such that: $x^2+y^2+xy=3$. Find the maximum of $x^2y$

hello, we get this here $$\left\{\frac{1}{3} \sqrt{2 \left(69-11 \sqrt{33}\right)},\left\{x\to \sqrt{\frac{1}{6} \left(15-\sqrt{33}\right)},y\to \frac{2 \sqrt{2 \left(69-11 \sqrt{33}\right)}}{15-\sqrt{33}}\right\}\right\}$$Sonnhard.
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bel.jad5
3750 posts
bel.jad5
#3 h  Sep 19, 2016, 8:14 PM • 2 Y
Y by Adventure10, Mango247
Yes true.
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ali3985
1042 posts
ali3985
#4 h  Sep 22, 2016, 9:39 PM • 3 Y
Y by bel.jad5, Adventure10, Mango247
bel.jad5 wrote:
Let x and y positive real numbers such that: $x^2+y^2+xy=3$. Find the maximum of $x^2y$

If we put $y=kx$, with k a positive real then

$x^2+y^2+y^2=x^2(k^2+k+1)=3$

$x^2y=kx^3=\frac{3\sqrt{3}k}{(k^2+k+1)^\frac{3}{2}}$

setting $f(k)=\frac{3\sqrt{3}k}{(k^2+k+1)^\frac{3}{2}}$

Then $f'(k)=\frac{3\sqrt{3}(-4k^2-k+2)}{(k^2+k+1)^\frac{5}{2}}$

$f'(k)=0 \iff k_{1}=\frac{\sqrt{33}-1}{8}$ or $k_{2}=\frac{-\sqrt{33}-1}{8} < 0 $

and we get

$f(k) \le f(k_{1})$
This post has been edited 1 time. Last edited by ali3985, Sep 22, 2016, 9:40 PM
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sqing
41238 posts
sqing
#5 h  Sep 24, 2016, 8:51 AM • 2 Y
Y by Adventure10, Mango247
http://www.artofproblemsolving.com/community/c4t243f4h1309979_inequality:
Let $x$ and $y$ be nonnegative real numbers such that $x+y+\sqrt{2x^2+2xy+3y^2}=4$.
Prove that$$x^2y\leq32\left(\sqrt{\frac{5}{3}}-1\right)^3$$
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sqing
41238 posts
sqing
#6 h  Sep 25, 2016, 12:01 AM • 2 Y
Y by Adventure10, Mango247
sqing wrote:
http://www.artofproblemsolving.com/community/c4t243f4h1309979_inequality:
Let $x$ and $y$ be nonnegative real numbers such that $x+y+\sqrt{2x^2+2xy+3y^2}=4$.
Prove that$$x^2y\leq32\left(\sqrt{\frac{5}{3}}-1\right)^3$$
Proof(elecaii1981):
Attachments:
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sqing
41238 posts
sqing
#7 h  Mar 20, 2025, 2:48 PM
Y by
Let $ a,b $ be nonnegative real numbers such that $ a^2+b^2+ab=3 . $ Prove that
$$ a^3b\leq \frac{ 13 \sqrt{13}-35}{3} $$Equality holds when $ a=\frac{\sqrt{11-\sqrt{13}}}{2} ,b= \frac{\sqrt{5-\sqrt{13}}}{2}. $
$$ a^2b\leq \frac{\sqrt{138-22 \sqrt{33}} }{3} $$Equality holds when $ a=\sqrt{\frac{15-\sqrt{33}}{6}},b=\sqrt{\frac{9-\sqrt{33}}{6}}. $
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